# Why do we have to dot product in the Low-rank Bilinear Pooling?

I was reading this paper Hadamard Product for Low-rank Bilinear Pooling. I understand what they are trying to say, but I don't know why we have to convert the element-wise multiplication into a scalar (using the dot product)

$$\mathbb{1}^{T}\left(\mathbf{U}_{i}^{T} \mathbf{x} \circ \mathbf{V}_{i}^{T} \mathbf{y}\right)+b_{i} \tag{2}\label{2}$$

Why do we have to multiply the resulting vector by the one vector? We would still use the multiplicative interaction between elements if we did not consider multiplying by that one vector.

$$f_{i}=\sum_{j=1}^{N} \sum_{k=1}^{M} w_{i j k} x_{j} y_{k}+b_{i}=\mathbf{x}^{T} \mathbf{W}_{i} \mathbf{y}+b_{i}\label{1}\tag{1}$$